# Questions tagged [complex-dynamics]

This tag is for questions relating to complex dynamics, study of dynamical systems defined by iteration of functions on complex number spaces. It was an area of research established by Fatou and Julia towards the beginning of the last century.

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### Solving $f(x) = f(\frac{a + b x}{c + d x}) = f(\frac{a' + b' x}{c' + d' x})$?

How to solve the equation $$f(x) = f(\frac{a x + b}{c x + d}) = f(\frac{a'x + b'}{c'x + d'})$$ For given real $a,a',b,b',c,c',d,d'$ ? Maybe this system of equations is a bit overdetermined in its ...
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### Where to find out more about escape time fractal made by $z_{n+1}=a^{z_n}$

I was messing around making a very very simple mandelbrot renderer in python, and decided to replace the z = z**2 + c in my code with ...
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### Is a boundary of a completely invariant set is still completely invariant under an analytic function?

$R(z)$ is a rational function from the extended plane to the extended plane. Consider $D$ is a union of components of the Fatou set of $R(z)$, If $D$ is completely invariant, then how to prove that ...
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### Sign of $n$ th derivative of $f(x)$?

Let $f(z)$ satisfy $f(f(z)) = \operatorname{arcsinh}(z/2)$ More precisely, we construct such an $f(z)$ by using the fixpoint at $0$ and the related Koenigs function. see : https://en.wikipedia.org/...
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### Introducing undergraduate students to dynamical systems

In my department a course on dynamical systems is offered this semester. It is a course offered to third (out of four) year undergraduate students and it involves basic dynamics of real maps, ...
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### Equicontinuous Family at a Point

I am reading the definition of equicontinuous family at a point from book called "Iteration of Rational Functions" by Alan F. Beardon. There it is written that equicontinuity of family at a ...
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### Construction of Mandelbrot Set

I am doing Master's project in Complex Dynamics. Here I want to talk particularly about the Mandelbrot set. I have studied its formation and dynamics as parameter $c$ changes (which is very hand ...
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### Hyperbolic Set of Extended Complex Plane

I am studying one of the research article "The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets" by Mitsuhiro Shishikura. There is one section where he gave the ...
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### Math function for plotting a surface that looks like crumpled paper?

Is there a way to draw a two-dimensional or three-dimensional surface that resembles this? All I know so far is that the dynamics of paper are more complicated than they sound, does this mean I could ...
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